Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

Transcription

1 Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods arxiv: v1 [cond-mat.str-el] 25 Nov 2016 Jutho Haegeman, 1 and Frank Verstraete 1,2 1 Ghent University, Faculty of Physics, Krijgslaan 281, 9000 Gent, Belgium 2 Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria Keywords Equilibrium and Non-Equilibrium Statistical Physics, Many-Body Physics, Quantum Spin Chains, Tensor Networks, Entanglement, Bethe Ansatz, Fusion Tensor Categories Abstract Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators. 1

2 Contents 1. Introduction Matrix Product Operators: Definitions and Normal Forms Matrix product operators and transfer matrices Equilibrium stastistical physics and counting problems Path Integrals in 1+1 D Matrix product operators in PEPS: physics at the edge Nonequilibrium statistical physics and cellular automata in 1D Exact methods for diagonalizing matrix product operators Mapping to free fermions The Bethe Ansatz Discrete MPO algebras and tensor fusion categories Non-equilibrium steady states as matrix product states Numerical methods for diagonalizing matrix product operators Manifold of uniform matrix product states and its tangent bundle Algorithms for obtaining normal forms of uniform matrix product states Approximating the MPO eigenvalue problem in the MPS manifold Algorithms for the leading eigenvector of a matrix product operator Algorithms for the excited states of a matrix product operator Examples Outlook and conclusion A. Explicit representation of the matrix product state steady state for the asymmetric exclusion process with parallel updates Introduction The theory of entanglement is providing a novel language by which quantum many body systems can be described. The entanglement features of quantum spin systems are most clearly expressed in the language of tensor networks, in which the wavefunction of a many body system is encoded in a series of tensors, one for each physical spin. As opposed to the usual treatments of many body systems, the tensor network language treats systems in real space, which makes them especially amenable for strongly interacting systems. The central premise of tensor networks is the fact that an efficient local description of relevant many body wavefunctions can be obtained by modelling the way in which entanglement and correlations are distributed. This means that instead of working with the original degrees of freedom, we want to describe all physics by working on the Hilbert space of entanglement degrees of freedom connecting a bipartition of the system. As those degrees of freedom live on an interface, the spatial dimensionality of this Hilbert space is reduced. Hence, e.g., a one-dimensional system spin chain can be described by an effective zerodimensional theory, and a two-dimensional theory by an effective one-dimensional Hilbert space. Tensor networks therefore provide an explicit representation for a research program originally laid out by Feynman [1]: Now in field theory, what s going on over here and what s going on over there and all over space is more or less the same. What do we have to keep track in our functional of all things going on over there while we are looking at the things that are going on over here?... Its really quite insane actually: we are trying to 2 Haegeman and Verstraete

3 find the energy by taking the expectation of an operator which is located here and we present ourselves with a functional which is dependent on everything all over the map. Thats something wrong. Maybe there is some way to surround the object, or the region where we want to calculate things, by a surface and describe what things are coming in across the surface. It tells us everything thats going on outside... I think it should be possible some day to describe field theory in some other way than with wave functions and amplitudes. It might be something like the density matrices where you concentrate on quantities in a given locality and in order to start to talk about it you don t immediately have to talk about what s going on everywhere else. Tensor networks aim to do exactly that, by defining a new Hilbert space on the surface. Other powerful methods for simulating strongly correlated systems can also be understood from the point of view of Feynman. For example, dynamical mean field theory [2] models the outside degrees of freedom as a system of free fermions in a self-consistent way; we will later see that such an approximation is exact in e.g. the case of the 2-dimensional classical Ising model. But the message is clear: if we want to describe strongly correlated systems in such a way that we do not encounter an exponential wall, we better try to model the entanglement degrees of freedom. The theory of quantum information, originally aimed at the study of how to harness the power of the quantum world for doing information theoretic tasks, has in recent years made huge progress in understanding and classifying entanglement [3]. A crucial insight has been that ground states of Hamiltonians with local interactions, or equivalently all states whose marginal reduced density matrices are extreme points in the set of all marginals of the set of wavefunctions with certain symmetries, exhibit very few entanglement [4]: quantum correlations are essentially restricted to be local, but in such a way that e.g. translational symmetry is not broken, and this implies the so-called area law for the entanglement entropy [5, 6, 7]. The structure of tensor networks is precisely based on that insight [8]: we consider a graph with an edge between any degrees of freedom which interact with each other through a Hamiltonian term, and then define tensors on the vertices whose indices are contracted according to the underlying graph. There are essentially three different classes of tensor networks. First of all, there are the ones in which the graph underlying the tensor contractions has no loops, such as matrix product states (MPS) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and tree tensor networks [21, 22]. Those are very well understood due to the fact that a normal form exists for them. Second, there are the tensor networks with loops, such as projected entangled pair states (PEPS) [8, 23, 24, 25]. Although such tensor networks could in principle be hard from the computational complexity point of view [26, 27], in practice efficient methods have been constructed to deal with them [8, 28, 29, 30, 31, 32]. The third class of tensor networks involves the ones with an additional dimension which plays the role of scale in a renormalization group approach; those are called MERA (Multiscale Entanglement Renormalization Ansatz) [33, 34]. All three of those classes can be shown to arise naturally from a compression of the path integral representation of the quantum state ψ lim τ exp( τh) Ω [35, 36]. A central object in all those tensor networks is played by so-called matrix product operators (MPO) [37, 38, 39, 40]. The aim of this paper is to present a review of the many facets of MPOs. Just as usual operators in quantum mechanics, MPOs come in several flavours. On the one hand, MPOs can represent density operators [such as exp( βh)]. On Exact and Computational Methods for Matrix Product Operators 3

4 the other hand, MPOs form a very convenient way of representing a multitude of interesting operators acting on the Hilbert space of interest such as Hamiltonians for quantum spin chains. Similarly, the transfer matrices in a path integral formulation are MPOs, just as cellular automata for non-equilibrium spin systems. Interesting combinatorial problems can also be rephrased as calculating partition functions using the transfer matrix technique. And MPOs also play a crucial role in two-dimensional PEPS, for which the matrix product operators act on the virtual space and encode how the quantum correlations in the system are distributed. Matrix product operators have since long been part of the canon of statistical physics, albeit typically under different nomenclatures. The most famous application of an MPO is probably Onsager s solution of the 2-dimensional Ising model [41]; Onsager managed to diagonalize the corresponding transfer matrix (which is a simple MPO) that was originally introduced by Kramers and Wannier [42]. The calculation of the entropy of 2-dimensional spin ice, a combinatorial problem, has been achieved by finding the largest eigenvector of an MPO [43]. Similarly, the central ingredient in solving integrable spin systems makes essential use of matrix product operators: the core of the algebraic Bethe ansatz [44, 45] is the construction of an algebra of matrix product operators. Matrix product operators can also be used to construct explicit representations of tensor fusion categories [46, 47, 48, 49, 50], which makes them central to the study of topologically ordered systems. MPOs have also been studied extensively in the context of non-equilibrium systems [51, 52], as one-dimensional cellular automata are MPOs of a special kind [53]. Finally, MPOs have become a very powerful tool for simulating quantum spin chains numerically, as the MPO formalism forms the backbone of the density matrix renormalization group (DMRG). During the last years, a lot of progress has been made both on the theoretical description of MPOs and on the computational aspect of diagonalizing them. One of the goals of this article is to discuss how those new developments fit within the many classic applications of MPOs. This article is based on a series of lectures that was given to master students at the university of Vienna in the winter semester 15-16, which explains the idiosyncratic choice of topics and examples and the fact that plenty of interesting and relevant topics have not been included. 2. Matrix Product Operators: Definitions and Normal Forms In this review, we focus on translational invariant matrix product operators. In particular, we are interested in constructing uniform families of MPOs which act on a tensor product of N d dimensional spins and hence on a Hilbert space of dimension d N. Note that this Hilbert space could equally well represent a quantum many body state ψ = ψ(i 1, i 2,..., i N ) i 1 i 2... i N, i 1 i 2...i N normalized with the 2-norm, a classical probability distribution p(i 1, i 2,..., i N ), as used in the case of cellular automata, or just a vector in a d N dimensional (real) Hilbert space, as occurs in transfer matrices of classical spin systems. Translational invariant Matrix Product Operators are defined by a single 4-leg tensor and A ij αβ (see Figure 1) where the upper indices act on the physical space 1 i, j d 4 Haegeman and Verstraete

5 Figure 1: Graphical notation of a tensor contraction and of a matrix product operator. while the lower ones are contracted (hereafter called virtual) and run from 1 to D, and by an additional matrix M αβ which dictates how the MPO has to be closed. Written out in components, we have Ô(A ij, M) = i 1 j 1 i 2 j 2... ( ) Tr A i 1j 1 A i 2j 2... A i N j N M i 1 j 1 i 2 j 2... i N j N. A sufficient (but not necessary) condition for translational invariance is that M commutes with all A: [M, A ij ] = 0. Permutational invariance is only ensured when all the A i commute with each other (and M can then be arbitrary). Alternatively, we can of course also parameterize MPOs with 3-leg tensors and a basis of local operators {O i} Ô = i 1 i 2... ( ) Tr A i 1 A i 2... A i N M O i1 O i2... O in which can be a more useful representation when constructing algebras of MPOs. This parameterization of MPOs is not unique, as a gauge transformation of the form A ij XA ij X 1, M XMX 1 leaves the MPO invariant. This is a purely virtual property, so that the discussion of the resulting canonical forms does not distinguish between the case of MPOs (where the matrices are labelled by a double index A ij ) or MPS (where they are labeled by a single index A i ). Both terminologies are used interchangeably. If the matrix multiplication algebra of the matrices A ij spans the complete D 2 dimensional matrix space, then the corresponding MPO is called injective. If not, then there must exist invariant subspaces P 1, P 2, such that i, j, r : A ij P r = P ra ij P r. We can always perform a unitary U gauge transform 1 to bring the P rs into block diagonal form, where the unitary is unique up to a direct sum of unitaries on the individual subspaces. Let us for example consider the case with 2 invariant subspaces: UA ij U = [ ] A ij 11 Aij 12 0 A ij 22 1 The fact that this can be done in a unitary way can be seen as follows: there must be a basis X such that all XA ij X 1 are upper block triangular. Hence the span of all matrices which are in the null space have mutually orthogonal row and column vectors, and we can choose U as the basis for those vectors. Exact and Computational Methods for Matrix Product Operators 5

6 Figure 2: The transfer matrix of a matrix product operator, its leading eigenvector, and a matrix product operator in the right canonical form. where A ij 11 and Aij 22 now represent injective MPOs. If furthermore [ ] UMU M 11 M 12 = 0 M 22 then the matrices A ij 12 and M12 do not affect at all the physical degrees of freedom of the MPO, and we can get rid of them (i.e., setting them equal to zero). In that case, the MPO is nothing but a direct sum of the two other MPOs Ô(Aij 11 ) and Ô(Aij 22 ) which can on their turn be injective or not. If not, we can repeat the same procedure. However, if M is not of the form quoted above (i.e. has a block M 21), then we cannot get rid of the degrees of freedom in A ij 12 ; this situation is used to encode e.g., a sum of local operators such as appearing in a Hamiltonian for a system with periodic boundary conditions. In the injective case and when M = I (which is then the only choice that commutes with all A ij ), it is easy to define a normal form. A crucial object of interest in that case is the MPS transfer matrix 2 E = ij Aij Āij (see Figure 2). This matrix defines a completely positive linear map: E ρ A ij ρa ij. ij The quantum version of Perron-Frobenius theory [54, 11] now dictates that injectivity (or ergodicity) implies that the eigenvector ρ corresponding to the largest eigenvalue λ > 0 of E is unique and can be chosen to be strictly positive in the semidefinite sense. Let us now do the gauge transform X = ρ 1/2 and a rescaling with factor λ 1/2. The ensuing à ij = λ 1/2 ρ 1/2 A ij ρ 1/2 then constitute an isometry (see figure 2). We could of course equally well have repeated this procedure starting from the left eigenvector, which would have resulted in an isometry with respect to the different indices. This procedure is called 2 Here we switch the discussion to MPS because the transfer matrix originates from interpreting the MPO Ô(A) as a state O(A) in a Hilbert space H H and applying the 2-norm. 6 Haegeman and Verstraete

7 Figure 3: Graphical proof of the fundamental theorem of matrix product states. bringing an injective MPS into right or left normal form, which is unique up to an additional unitary gauge transform. Let us now prove a very useful theorem, which we call the fundamental theorem of MPS, on the uniqueness of the MPS normal form for translational invariant systems [55]: Theorem 1 Given two injective MPS characterized by A ij and B ij, then Ô(Aij ) = Ô(Bij ) for N if and only if there exists a gauge transform which transforms A ij into B ij : A ij X = XB ij. This can readily be proven by using the Cauchy-Schwarz theorem. Without loss of generality, we can assume that both tensors are in right normal form, and we furthermore assume that the largest eigenvalue in magnitude of their respective transfer matrices is equal to 1 (which can always be obtained by rescaling the tensors with a constant). Then Ô(A ij ) = Ô(Bij ) iff O(A) O(B) = 1, which is possible iff the largest eigenvalue of the mixed transfer matrix ij Aij B ij has magnitude 1. Let us call the associated eigenvector X, and then use the Cauchy-Schwarz theorem with respect to the dotted line in Figure 3. Because the inequality must be an equality, we conclude that XB i = A i X which we set out to prove. We are now in a position to define the normal form of an MPO [18, 56]: Given a translational invariant MPO {A ij, M}, then its normal form is obtained by first identifying its invariant subspaces and unitary gauging all A ij in upper triangular block form A ij ab. We can also write M in this basis, and write it as M = M 1 + M 2, where M 1 is block upper triangular and M 2 only has blocks beneath this block diagonal. The complete MPO is a sum of O 1{A ij, M 1} and O 2{A ij, M 2}. Two MPOs will be equal to each other iff both of those blocks are equal to each other. For O 1, we get rid of the off-diagonal blocks in A ij and M 1, and if the remaining diagonal blocks are injective, we are done. Otherwise, we have to find the invariant subspaces within those blocks and repeat the same procedure. Potentially, the same injective MPS can appear multiple times, and if this is the case, we make one block out of it and put the right weights into the corresponding block of M 1 by multiplying it with an appropriate scalar. In case 2, translational invariance is difficult to judge, as it could be obtained without all blocks in A ij on which M 2 has support to be equal to each other. See [18, 56] for a proof of uniqueness. To conclude this section, let us give a few examples. Let us consider the projector on the subspace of all qubit strings with even parity P = ( I + Z N ) /2. This can readily be written as an MPO with bond dimension 2 where Exact and Computational Methods for Matrix Product Operators 7

8 both physical and virtual indices take value in Z 2 = {0, 1}: A ij αβ = δijδ αβ ( 1) i α and [ ] [ ] α [ ] 1/2 0 M =. Equivalently, we can write A ij αβ 0 1/2 = Note that this MPO is non-injective and consists of 2 injective blocks with bond dimension 1. A less trivial example, illustrating the existence of so-called trash vectors in the reduction to the canonical form, is encountered in the context of symmetry protected topological (SPT) phases. Let us consider the so-called CZX [47] MPO A ij [ ] αβ = δ βi ( 1) α i 0 1 for i, j, α, β = 0, 1. This MPO is injective, as can easily be seen by 1 0 ij [ ] [ ] considering the algebra generated by the matrices A = and A = Let us now square this MPO, thereby getting an MPO B with bond dimension 4: [ ] [ ] B = = [ ] [ ] B = = This MPO is clearly not injective, and it has an invariant subspace P = We can therefore as well work with the block [ ] B =, B 11 = 0 0 [ ] But[ this MPO ] again has an invariant subspace given by the projector P = 1 1 1/2. We can rotate the MPO in this basis, get rid of the trash vector, 1 1 and end up with an MPO with bond dimension 1. In summary, the normal form of the MPO Ô(A)2 is B ij = ( 1)δ ij, which globally means Ô(A)2 = ( 1) N I. Finally, let us construct the MPO for the operator Ô = i Xi. It can readily be checked that the MPO [ ] [ ] [ ] [ ] A ij αβ = ij αβ [ ] 0 0 together with M = does the job. Clearly, A is not injective, but as M is of 1 0 the form M 2, it is not possible to reduce this form further. ij ij αβ αβ 8 Haegeman and Verstraete

9 3. Matrix product operators and transfer matrices Matrix product operators originally popped up as transfer matrices in statistical physics, and more specifically in the study of partition functions of classical spin systems. However, such transfer matrices can also be defined from the path integral representation of quantum spin systems. In a similar fashion, the transfer matrix can be introduced for tensor network representations of quantum states, where it often plays an important role in formulating the associated tensor network algorithms. Finally, we discuss the role of matrix product operators in the context of nonequilibrium statistical physics Equilibrium stastistical physics and counting problems he most famous application of the transfer matrix method in classical statistical physics is for sure Onsager s exact solution of the 2-dimensional Ising model, which was obtained by diagonalizing its transfer matrix, but many other fascinating partition functions have been diagonalized using the more advanced Bethe ansatz. In particular, combinatorial problems on the lattice can often also be formulated as the zero temperature entropy of a partition function and can then be determined using the transfer matrix method Combinatorial problems on the lattice. Counting problems on lattices are notoriously hard and can exhibit very rich behaviour. The prime example of such a problem is calculating the scaling of the number of ways in which a given set of tiles can tile the (infinite) plane. Such problems have a very natural formulation in terms of the leading eigenvalues of a transfer matrix. Before we move to the interesting case of 2-dimensional lattices, let us warm up with the simple task first undertaken by Fibonacci of counting the number of ways in which we can arrange N bits such that a bit 1 is never followed by another 1. We can easily set up a recursion for this: let us call the number of allowed sequences of n bits ending with a 0 as x(n), and the number of allowed sequences ending with a 1 as y(n). Then we trivially obtain: [ ] [ ] [ ] [ ] x(n + 1) 1 1 x(n) = = T n x(1) y(n + 1) 1 0 y(n) y(1) }{{} T The relevant (exponential) scaling is governed by the leading eigenvalue of the matrix T, which is equal to (1 + ([ ] ) 1 1 5)/2. The exact result is given by Tr T N. For the 1 1 case of N bits on a ring, the result is just TrT N = [ (1 + 5) N + (1 5) ] N /2 N. Let us investigate whether it is possible to write the projector on those allowed sequences as a matrix product operator. The MPO Ô defined by the following tensor does exactly that: [ ] αβ = 1 1 δiαδij 1 0 A ij We indeed recover the counting result by taking the trace of Ô. The situation is much more interesting however in two dimensions. Let us stick to this Fibonacci example but try to count the ways in which I can put bits on a square lattice with N M sites in such a way that a bit 1 is solely surrounded by 0s. This MPO can be constructed by constructing an αβ Exact and Computational Methods for Matrix Product Operators 9

10 MPO with periodic boundary conditions on N sites : [ ] [ ] A ij αβ = δiα The result is then obtained by calculating Tr ( ˆ O N M ). In the limit of large M, this amounts to calculating the leading eigenvalue of the MPO Ô. We will see in later sections how such problems can be solved to a very good precision using tensor network techniques. A slight modification of this problem is obtained by adjusting the weight of the occupied sites (i.e. 1s): [ ] [ ] [ ] A ij αβ = p iα By varying p from 0 to, we can adjust for the density of occupied sites from 0 to 1/2, and can count the number of possible solutions with the corresponding density: in the thermodynamic limit, the law of large numbers dictates that the weight of the partition function will be very peaked around solutions equal to the average density ρ, and hence the total number of solutions can be obtained by calculating Z/p ρnm or exponential scaling 1 NM log(z) ρ log(p). Note that ρ itself can either be calculated as an expectation value in the tensor network, or as a derivative of the partition function with respect to p. This model turns out to exhibit a nontrivial phase transition at a finite density, and such a phase transition is called a jamming transition. The history of such transitions goes back a long time to the birth of the famous Metropolis sampling algorithm, in which Rosenbluth et al. tried to calculate the entropy of hard disks [57]. The MPO discussed here yields a discretized version of this hard disk problem, in which hard disks are replaced by occupied sites which cannot be adjacent to other occupied sites. If we would have considered a triangular lattice as opposed to a square one, this problem is equivalent to solving the hard hexagon 3 problem, which has been solved exactly by Baxter using an ingenious combination of the Bethe ansatz and corner transfer matrix ideas [58]. Given a square lattice, another classic problem is to count the number of ways in which we can fill the lattice with dimer configurations. This problem is equivalent to counting the number of ways in which we can label the edges of a square lattice with a bit 0 or 1 in such a way that around any vertex there is precisely one edge equal to 1 and three of them are equal to zero. Let us therefore consider the tensor A ij αβ equal to one iff one of its indices is one and equal to zero otherwise. If we now construct the partition function out of this tensor by putting it on any vertex of the square lattice and contract this two-dimensional tensor network, we will count the number of possible configurations. From the point of view of MPOs, this partition function is equal to Z = lim N Tr (ÔN ) N. The relevant quantity is log(z)/n 2, as this yields the exponential scaling of the number of configurations per lattice site. It turns out that this problem is exactly solvable [59]. Within the computer science community, the preferred method for finding an exact solution is in terms of Pfaffians. Physicists prefer to tackle it using the very related method of reinterpreting the transfer matrix as a thermal state of free fermions, which can be diagonalized exactly [60]. This will be discussed in Section 4. ij ij αβ αβ 3 Note that the hexagonal lattice is the dual lattice of the triangular lattice 10 Haegeman and Verstraete

11 Another classic counting problem was originally formulated by Pauling [61], where he was able to explain the experimentally determined zero entropy of ice at absolute zero by counting the possible ways in which two of the four vertices of tetrahedra tiling the hexagonal wurtzite crystal are occupied. Solving this 3-dimensional problem would in principle involve finding the largest eigenvalue of a projected entangled pair operator (PEPO), which is the two-dimensional generalization of the one-dimensional MPO. The two-dimensional version of the problem was formulated by Lieb [43], and is as follows: given a square lattice, put degrees of freedom s ij {, } on the edges. Count in how many ways we can label the edges such that around every vertex there are exactly two spins up and two spins down. The transfer matrix MPO for this problem can readily be written down: A ij αβ = { 1, i + j + α + β = 2 0, i + j + α + β 2 For obvious reasons, this model is also called the six vertex model, and it turns out that this transfer matrix can be diagonalized exactly using the Bethe ansatz. The Bethe ansatz itself can be formulated in an algebraic way and involves a very nice application of matrix product operator algebras [45] Equilibrium Statistical physics in 2D. The central quantity of interest in the context of statistical physics is the partition function Z: given a system of N classical spins {s i} and an energy functional H(s 1, s 2, ), then the partition function is given by Z = e βf = e βh(s 1,s 2, ). s 1 s 2 where F = E T S is the free energy which is an extensive thermodynamic quantity. For infinite systems and Hamiltonians with local interactions (such as already encountered in the counting problems), this free energy is proportional to the logarithm of the leading eigenvalue of a transfer matrix / MPO with bond dimension equal to the local spin dimension. Indeed, let us illustrate this for the case of a translational invariant 2-body nearest neighbour Hamiltonian acting on d level systems. Define the matrix G G ij = exp ( H(s i, s j)/t ) then we can parameterize the transfer matrix by the MPO A ij αβ = δiαgijg αβ For the case of the ferromagnetic Ising model, G is given by G = [ e β e β e β e β The problem of diagonalizing such transfer matrices is equivalent to the problem of diagonalizing Hamiltonians of quantum spin systems, which is notoriously hard. However, beginning with the density matrix renormalization group (DMRG) [12, 9, 15] a wealth of methods has recently been developed for tackling that problem, which effectively allows one to find the leading eigenvalues and eigenvectors. These methods are effectively variational methods within the manifold of matrix product states, and we will discuss this framework in Section 5. These methods allow one to simulate local classical spin systems in two spatial dimensions to almost machine precision. ]. Exact and Computational Methods for Matrix Product Operators 11

12 The Ising model was originally introduced by Lenz as a toy model for explaining ferromagnetism and its related phase transition from a disordered phase to an ordered one. The first breakthrough happened when Kramers and Wannier wrote down the partition function of the two-dimensional Ising model in terms of the transfer matrix [42]. They realized that the Ising model is self dual. This means that, by a change of variables, the partition function at high temperature can be mapped to the one at low temperature. This way they could exactly identify the critical temperature, under the condition that there is a phase transition going on. Onsager s exact solution of this model in two dimensions is without a doubt one of the highlights of 20 th century statistical physics. The exact solution forced the whole community to rethink the whole basic formalism of second order phase transitions, and this was the starting point for studying critical phenomena and culminated in the formulation of the renormalization group. The reason for that upheaval is the fact that the exact critical exponents did not coincide at all with the mean field ones as were predicted. Onsager s solution was obtained by diagonalizing the transfer matrix introduced above; it turns out that it can both be diagonalized using the Bethe ansatz or using the Jordan Wigner transformation as discussed in the case of dimers Path Integrals in 1+1 D An extremely powerful way of dealing with quantum many body systems is given by the path integral approach. It also forms the basis of quantum Monte Carlo methods [62, 63]. The objective of the path integral formalism is to calculate local expectation values with respect to thermal states of interacting Hamiltonians. This is obtained by mapping a 1D quantum problem to an effective 2D classical-like partition function. The main idea is to introduce resolutions of the identity within the Gibbs state exp ( βh) as e βh = x 1 x 1 x 1 e β N H x 2 x 2 e β N.H x 3 x N 1 e β N.H x N x N x 2 x N If the resolutions of the identity have a tensor product structure ( x x(1) x(2) x(3) ), then ˆT = x i e β N H x j is in the form of a matrix product operator, albeit possibly with a very large bond dimension. The ground state of the Hamiltonian H is then nothing else than the leading eigenvector of this transfer matrix. In principle, is is very hard to deal with this transfer matrix T = x i e β N H x j as H consists of non-commuting operators. However, for N 1, a very good approximation of ˆT = x i e β N H x j can readily be found using the Trotter-Suzuki formula. Let us e.g. consider the Heisenberg antiferromagnetic spin chain with Hamiltonian H heis = i X ix i+1 + Y iy i+1 + Z iz i+1 where X, Y, Z represent the spin 1/2 Pauli matrices. Then a natural choice for the Trotter-Suzuki formula is to split the Hamiltonian into the terms that act on the even-odd sites and the odd-even ones: exp( ɛh) i ( ɛĥ2i,2i+1 ) exp }{{} exp( ɛ)(i+(exp(4ɛ) 1) ψ ψ ) i ( ɛĥ2i 1,2i ) exp 12 Haegeman and Verstraete

13 Figure 4: The tensor A ij αβ for which the leading eigenvector of the corresponding matrix product operator is the ground state of the Heisenberg Hamiltonian. The leading eigenvector of this MPO will be very close to the ground state of the Heisenberg model. For this particular example, we can rotate every second spin with the unitary operator Z, rotating every ψ into ψ +. The ensuing matrix product operator only has positive elements, and we can hence interpret is as a classical statistical mechanical model with only positive Boltzmann weights. This makes it possible to use classical Monte Carlo sampling techniques to simulate this model, and this is called quantum Monte Carlo. However, the fact that we could make all weights in the MPO positive is certainly an accident, and it is unknown how to achieve this for many of the most interesting problems; this is precisely the infamous sign problem [64]. Note that there are many ways in which we can split our Hamiltonian within the Trotter- Suzuki splitting. The one just discussed breaks translational invariance, as the even and odd sites were treated on an unequal basis. It is also possible to keep translational invariance at the cost of breaking the SU(2) invariance by splitting the Hamiltonian into three parts, containing all X, Y and Z terms. As shown in [39], an exact representation of exp ( x ) i ZiZi+1 can be obtained in terms of a MPO with bond dimension 2: exp ( α i Z iz i+1 ) = k 1 k 2 ( ) Tr C k 1 C k2 Z k 1 1 Z k 2 2 where C 0 = C 1 = [ ] cosh(α) 0 0 sinh(α) [ ] 0 sinh(α) cosh(α) sinh(α) cosh(α) 0 The formulas for the Pauli X and Y terms are of course equivalent. For the particular case of the ground state of the one dimensional Heisenberg model, the exact ground state can actually be found as the leading eigenvector of the transfer matrix associated to a 6-vertex model. This MPO has bond dimension 2 and is equal to A ij αβ = δijδ αβ 1 2 δiαδ jβ, as shown in figure Matrix product operators in PEPS: physics at the edge Projected entangled pair states (PEPS) [8, 23, 24, 25] provide a systematic way of parameterizing ground state wavefunctions of strongly correlated systems on two-dimensional lattices. PEPS form the natural generalization of matrix product states to higher dimensions. The archetypical example of a PEPS is the wavefunction of Affleck-Kennedy-Tasaki-Lieb on the square lattice [10], which is a spin-2 antiferromagnet obtained by projecting halves of four virtual singlet pairs onto the spin-2 subspace. More generally, a translation invariant Exact and Computational Methods for Matrix Product Operators 13

14 Figure 5: Definition of a projected entangled pair state parmaterized by the tensor A i αβγδ. PEPS on a square lattice is parameterized by a tensor A i αβγδ, with i the physical dimension and α, β, γ, δ virtual indices having dimension D; see figure PEPS transfer matrix. The complexity of calculating expectation values of observables is equal to the complexity of calculating expectation values of classical spin systems in 2D. Indeed, the quantity ψ O ψ involves the contraction of a tensor network in which the bra and the ket are connected to each other. Just as the central interest in 2D classical systems is contained in the eigenstructure of the transfer matrix, here the essential ingredients are encoded into the leading eigenvector of the double layer transfer matrix (see Figure 6), which is a vector in a Hilbert space of dimension D 2N. Due to the symmetric double layer structure of the transfer matrix, it makes however more sense to interpret this transfer matrix as a completely positive map on operators of dimension D N D N. The theory of completely positive maps (quantum Frobenius theorem [54]) then dictates that the leading eigenvector can be chosen to be a positive operator ρ. The quantum analogue of ergodicity is called injectivity, and injectivity ensures the uniqueness of this fixed point [65]. From the computational point of view, the class of PEPS seems to be a very rich variational class by which it should be possible to describe the physics of many gapped phases of matter. The central difficulty and bottleneck in implementing the variational optimization is exactly the step of finding the leading eigenvector of the transfer matrix, and the motivation of coming up with better algorithms for finding fixed points of MPOs stems to a large extent from this problem Entanglement Hamiltonians. As the leading eigenvector of the transfer matrix can be interpreted as a hermitian operator itself, it has itself eigenvalues and eigenvectors. From 14 Haegeman and Verstraete

15 Figure 6: Definition of a projected entangled pair state parameterized by the tensor A i αβγδ. the point of view of the theory of entanglement, those eigenvalues are of great interest as they represent the so-called entanglement spectrum [66] and their entropy is the entanglement entropy which exhibits the famous area law [7]. Much of the recent progress in understanding exotic and strongly correlated phases of matter stems from studying the structure of those eigenvalues. The framework of tensor networks is very well suited for studying this entanglement spectrum in more detail [67]. As opposed to other methods, tensor networks also give access to the eigenvectors corresponding to the entanglement spectrum, which is especially interesting from the point of view of topological phases where one might study how those eigenvectors transform under certain symmetries. As shown in Section 5, it is possible to find the leading eigenvector of the double-layer transfer matrix of the PEPS in terms of an MPS of a certain bond dimension χ, and as just discussed, this MPS can effectively be interpreted as an MPO of the same bond dimension. As the double layer transfer matrix is typically not hermitian, the left and right eigenvectors are typically different. Let us call the corresponding MPOs ρ L and ρ R. Without loss of generality, we can always rescale ρ L, ρ R such that they are hermitian and positive. In analogy to the way in which the entanglement spectrum is determined in 1D MPS, the entanglement spectrum is obtained by calculating the singular value decomposition of the operator ρ L ρr, or equivalently the eigenvalue decomposition of ρ Lρ R ρl, the logarithm of which we call the entanglement Hamiltonian. Note that this operator has the same eigenvalues as the operator ρ Lρ R, which is the product of two MPOs and hence an MPO itself. The entanglement spectrum can be obtained by finding the eigenvectors and eigenvalues of this MPO. We will show that numerical techniques based on MPS allow to do exactly that Matrix product operators and topological order. PEPS turn out to be also very useful for characterizing systems with topological quantum order. In that case, PEPS exhibit nontrivial symmetries on the virtual level [68, 69, 48, 50]. Such systems with topological order are currently under intense study as they present a natural framework for constructing quantum error correcting codes which allow to build fault tolerant quantum memories and quantum computers. The simplest of those systems is the so-called toric code [70], originally introduced by Kitaev to exemplify the connection between quantum error correcting codes and systems Exact and Computational Methods for Matrix Product Operators 15

16 Figure 7: For a topologically ordered projected entangled-pair state, the symmetry on the virtual level is characterized by a set of matrix product operators (dotted). exhibiting topological quantum order and anyonic excitations. By blocking 4 sites into 1, one can parameterize one of the ground states by a translational invariant PEPS with bond dimension 2 [26, 68]: A ijkl αβγδ = δi j,αδ j k,βδ k l,γ δ l i,δ with addition modulo 2. It can easily be seen that this PEPS is not injective, i.e. if we look at the tensor as a matrix from the physical level to the virtual one, then the matrix is not full rank. The full subspace on which the virtual system has support is in this case parameterized by the projector P 4 = 1 ( I + Z 4 ) 2 Here Z is the Pauli Z spin 1/2 matrix. Note that this is a very simple MPO with bond dimension 2 which we encountered in the examples of Section 2: A ij αβ = δ αβ(z ij) α. Note also that the same MPO characterizes the virtual subspace that a larger block of spins has support on. Let us indeed consider a contiguous block of spins on the lattice, and consider the entanglement entropy of this block. A well established property of topological systems is the fact that this entanglement entropy scales like the length L of the boundary of this block (so-called area law ), and that there is a constant correction γ to this area law which characterizes the anyonic content of it: S(ρ L) = cl γ [71, 72]. This γ turns out to be exactly related to the the non-injectivity of the PEPS: it arises as a consequence of the hidden symmetry on the virtual level of the PEPS. This hidden symmetry is characterized by a set of MPOs (see figure 7), and the full richness of the topological theory (including e.g. anyon statistics and braiding properties) can be deduced from studying the algebra generated by those MPOs [48, 50]. The so-called quantum double models and string nets [46] provide systems for which those MPOs can be constructed analytically. In the case of the toric code discussed before, those MPOs are obtained by bringing the MPO defined before into normal form, which is trivial in this case as it is the sum of 2 MPOs Ôi of bond dimension 1: Ô 0 is the identity matrix and Ô1 the tensor product of Zs. The ensuing algebra is in this case given by ÔiÔj = Ôi j. The way in which those symmetries are encoded in the PEPS tensors can be most easily understood in terms of the pulling through condition, pictorially given in figure 8; those MPOs can be interpreted as Wilson loops acting on the virtual level, but opposed to Wilson loops, they remain exact symmetries even in the presence of a finite correlation length. A consequence of this pulling through condition is the fact that the different MPOs commute 16 Haegeman and Verstraete

17 = =, Figure 8: states. The local pulling through condition in topological projected entangled-pair with the double layer transfer matrix: ] [Ôi Ôj, T = 0 Those symmetries are of course reflected in its leading eigenvectors, and the way in which those symmetries are broken characterize the different ways by which the different anyons in the theory can condense [73, 74]. Conversely, it is possible to try to characterize all possible sets of MPOs which form a closed algebra. Given such MPOs, it is possible to explicitly construct PEPS wavefunctions for which those MPOs represent their symmetries. The resulting classes of topological systems turn out to be precisely those proposed by Kitaev (quantum doubles [70]) and Levin and Wen (stringnets [46]) in the case of spin systems, but certainly allow for more general systems such as the ones consisting of fermions [75] and/or symmetry protected and symmetry enhanced topological phases [76] Nonequilibrium statistical physics and cellular automata in 1D Transfer matrices also naturally occur in the context of cellular automata and nonequilibrium physics of one-dimensional systems [see e.g. [53]]. In this context the MPO parameterizes a stochastic matrix which maps a probability distribution defined on a 1- dimensional line of d-level systems and hence of dimension d N to another probability distribution. If this stochastic matrix is ergodic, then Perron-Frobenius theory imposes that it has a unique eigenvector with eigenvalue 1, and this fixed point probability distribution plays the role of the stationary distribution under the stochastic updates Directed Percolation. As archetypical example, let us consider the problem of directed percolation: imagine an infinite square lattice, and erase any edge randomly with probability p; what is the probability that a connected path will exist from one side to the other side? This problem is an idealization of asking the question whether water can percolate through a rock with a certain density of cracks. The so-called Domany-Kinzel cellular automaton [53] can be considered as the Ising model of this class of percolation problems. Their cellular automaton acts on N bits x i, with x i = 1 if there is a particle present and 0 otherwise. It acts successively on all even and then on all odd sites: For even times: x 2k (t + 1 = 0 if x 2k 1 (t) + x 2k+1 (t) = 0 x 2k (t + 1) = 1 with probality p 1 if x 2k 1 (t) + x 2k+1 (t) = 1 and otherwise x 2k (t + 1) = 0 x 2k (t + 1) = 1 with probality p 2 if x 2k 1 (t) + x 2k+1 (t) = 2 and otherwise x 2k (2t + 1) = 0 Exact and Computational Methods for Matrix Product Operators 17

18 Figure 9: Matrix product operator representation of the cellular automaton for directed percolation. For odd times: x 2k+1 (t + 1) = 0 if x 2k (t) + x 2k+2 (t) = 0 x 2k+1 (t + 1) = 1 with probality p 1 if x 2k (t) + x 2k+2 (t) = 1 and otherwise x 2k+1 (t + 1) = 0 x 2k+1 (t + 1) = 1 with probality p 2 if x 2k (t) + x 2k+2 (t) = 2 and otherwise x 2k+1 (t + 1) = 0 The physics is contained in the eigenstructure of the transfer matrix MPO for 2 time steps. It has bond dimension 4 and is most easily depicted as in figure 9. Here the different tensors are G = and [ ] 1 1 p 1 1 p 1 1 p2 S = 0 p 1 p 1 p 2 The whole MPO is a stochastic matrix. The model exhibits an interesting phase diagram exhibiting second order phase transitions for a 1-parameter family p 1(c), p 2(c). Those phase transitions are of the class of directed percolation. However, this stochastic matrix is not ergodic, which means that it has several fixed points. Although matrix product state methods have been succesfully used for finding the leading eigenvectors of the continuous time version of this problem (where the MPO becomes a Liouvillian which is a sum of local terms), diagonalizing the above translational MPO in the thermodynamic limit using MPS methods remains a challenge, mainly due to the fact that a trivial (zero-entanglement) fixed point exists. This should be related to the complications which arise in the description of the universal behaviour of the percolation phase transition in terms of non-unitary conformal field theories Asymmetric Exclusion Processes. A different setting where stochastic cellular automata have been used to a great success is in the field of transport phenomena. Let us consider a line of N bits, and model traffic as a hopping process of particles from one site 18 Haegeman and Verstraete

19 Figure 10: Matrix product operator representation of the cellular automaton governing the asymmetric simple exclusion process with fully parallel updates. to the one to the right of it: this is called the asymmetric simple exclusion process (ASEP) [77]. More specifically, at every time step, a particle has probability p of moving to the right on the condition that there is no particle to the right of it. Furthermore, there is a constant inflow of particles on site 1 with probability p in and outflow of particles at site N with probability p out. This model exhibits an interesting phase transition as a function of these probabilities which many people experience on a daily basis: if the inflow becomes too large, the traffic gets stuck! In the literature, this model is called the ASEP with fully parallel updates. The MPO of this process can readily be constructed, see figure 10. Remarkably, an analytical form can be found for the leading eigenvector of this matrix product operator in terms of matrix product states [78, 79]. The essential ingredient of this construction is an explicit construction of a matrix representation of an algebra of operators [51, 52]. This will be discussed in the section on exact solutions. 4. Exact methods for diagonalizing matrix product operators There exists a vast literature in mathematical physics on exact solutions of eigenvalues and eigenvectors of transfer matrices and MPOs, and most of those exact solutions concern equilibrium and non-equilibrium statistical physics. In this article, we will only be able to discuss a few remarkable results. This section is divided in 4 parts. The first part discusses the mapping of a class of transfer matrices to Gibbs states of quadratic Hamiltonians of free fermions. This method allows for a huge simplification into diagonalizing the transfer matrix of the classical Ising model in 2D, as originally done by Onsager. As this proof is very well known, we present here the exact solution of another fascinating problem, namely the dimer problem on the square lattice. The partition function of that model was originally solved using Pfaffian methods by Kasteleyn [59], but here we follow Lieb [60] and show how a mapping to free Exact and Computational Methods for Matrix Product Operators 19

20 fermions allows to diagonalize the transfer matrix exactly. In a second part, we discuss the algebraic Bethe ansatz, as originally formulated by Faddeev and collaborators [44, 45]. A central role in integrable models solvable by the Bethe ansatz is played by an algebra of commuting matrix product operators. We show how an application of the fundamental theorem of MPOs to the associativity of this algebra immediately leads to the so-called Yang Baxter equation, and hence allows to reproduce classic results in the field of the algebraic Bethe ansatz in a very concise way. As an application of that formalism, we show how to diagonalize the transfer matrix of the 6 vertex model, and show how those eigenvectors are also the eigenvectors of the Heisenberg XXZ model. We also show how the coordinate Bethe ansatz can be formulated in terms of MPS, and how that representation is related to the one of the algebraic Bethe ansatz by a gauge transformation [80]. In a third part, we apply the logic as the one presented in the part on the algebraic Bethe ansatz, but now on a discrete finite set of MPOs fulfulling a nontrivial closed algebra. This allows us to characterize a large class of topological phases in 2D on the hand of matrix product operators. In the last part, we discuss the case of finding the leading eigenvectors of matrix product operators corresponding to nonequilibrium systems. This work was pioneered by B. Derrida and collaborators [51], and the central tool is to reformulate exact eigenstates in terms of matrix product states with infinite virtual bond dimension. It is then shown that the associated matrices form a very simple algebra, and the matrices satisfying this algebra have a very peculiar form. A central role is again played by the fundamental theorem of matrix product states Mapping to free fermions The most famous method for diagonalizing MPOs involves the mapping of the transfer matrix to a Gibbs state of free fermions; this happens to be possible for quite some interesting models. We illustrate this method by diagonalizing the MPO corresponding to the dimer problem on the square lattice. Recall that the MPO Ô was given by the tensor Aij αβ, which was equal to 1 iff one of its indices is one and equal to zero otherwise. From a conceptual point of view, the only thing we do is a basis change such that Ô becomes a tensor product of 4 4 matrices. We follow Lieb [60] in the following steps: It can readily be checked that the MPO can be rewritten as ( N ) ( N ) Ô = X n exp σn σ n+1 n=1 [ ] [ ] 0 1 where X = and σ 0 1 =. Note that this works because σ is nilpotent and that all terms σn σ n+1 commute. We now take the square of this MPO, which obviously has the eigenvalues squared: (Ô) N ( 2 N ) N ( N ) = X n exp σn σ n+1 X n exp σn σ n+1 n=1 n=1 n=1 n=1 ( N ) ( N ) = exp σ n + σ + n+1 exp σn σ n+1 n=1 n=1 n=1 20 Haegeman and Verstraete

21 Next, we define operators which are fermionic using the Jordan Wigner prescription: n 1 ψ n = ( Z m)σn m=1 Those operators and their hermitian conjugates obey the algebra of fermionic annihilation operators {ψ m, ψ n} + = δ mn and {ψ m, ψ n} + = 0. The important point is that this algebra is invariant under canonical transformations. One can check that Ô2 can be rewritten in terms of quadratic forms: ( ) ( Ô 2 = exp ψ nψ n+1 exp ) ψ nψ n+1 n n We then perform a canonical transformation of the fermionic operators to momentum space ψ n = 1 exp(ikn)χ k N where k ranges from π + 2π/N to π. We now have ψ nψ n+1 = 1 e ikn+ik (n+1) χ k χ k = e ik χ k χ k N n n,k,k k ψ nψ n+1 = 1 e ikn ik (n+1) χ k N χ k = e ik χ k χ k n n,k,k k k Due to e ik χ k χ k + e ik χ k χ k = 2i sin(k)χ k χ k and the fact that pairs of distinct fermionic operators commmute, we get Ô 2 = π(1 2/N) k=2π/n ( exp ) exp (2i sin(k)χ k χ k ) 2i sin(k)χ k χ k }{{} M k All matrices M k commute with each other, so the largest eigenvalue of our MPO is just the product of the largest eigenvalue of all of them. M k can be written out in spin components (inverse Jordan Wigner on 2 modes): i sin(k) M k = i sin(k) sin 2 (k) The largest eigenvalue is readily obtained as λ max(k) = ( sin(k) sin 2 (k) The leading scaling term in the partition function is therefore (where we include an extra factor 1/2 as we calculated the eigenvalues of Ô2 ): N log Z = 1 2N 1 2π π(1 2/N) k=2π/n π 0 dk log log λ max(k) ) 2 ( ) sin(k) sin 2 (k) Exact and Computational Methods for Matrix Product Operators 21

22 So the number of dimer configurations grows as ( ) NM. This classic result was originally proven by Kasteleyn using Pfaffians. Note that this method allows for finding all eigenvectors of the transfer matrix. This diagonalization of the transfer matrix of the dimer model automatically leads to an MPO description of all eigenstates: following [81], any canonical transformation of free fermions can be implemented exactly using a quantum circuit on spin 1/2s with only nearest neighbour interactions and of depth N 2 with N the number of spins involved. This automatically yields an MPO description, as any quantum circuit with nearest neigbour interactions is of course an MPO of a very special kind. Alternatively, we can use the matrix product operator representation of the Bethe ansatz to describe wavefunctions with fermionic statistics. The exact solution of the classical Ising model can be obtained along similar lines [82]. There is however an interesting complication in the symmetry broken regime, which manifests itself as a degeneracy of the leading eigenvector. To determine the order parameter in this symmetry broken state, one has to do perturbation theory with respect to an external magnetic field within this twofold degenerate subspace, and this leads to the famous magnetization formula of Onsager and Yang M T c T 1/8 [83] The Bethe Ansatz We first discuss the algebraic Bethe ansatz, as originally formulated by Faddeev and collaborators [44, 45], in terms of an algebra of commuting MPOs. As an application, we diagonalize the transfer matrix of the 6 vertex model, and show how those eigenvectors are also the eigenvectors of the Heisenberg XXZ model. Finally, we show how the coordinate Bethe ansatz, originally formulated by Hans Bethe [84], can be formulated in terms of MPS The algebraic Bethe ansatz. Matrix product operators are also at center stage in the algebraic Bethe ansatz, which has allowed for the solution of a multitude of models in statistical physics. The starting point is a continuous one-parameter family of MPOs with periodic boundary conditions, with the special property that any two of those MPOs commute with each other: λ, µ : [O(λ), O(µ)] = 0. The fundamental theorem of MPOs dictates that two (injective) MPOs (in this case O(λ)O(µ) and O(µ)O(λ)) are equal to each other if and only if there exists a gauge transform which transforms one MPO into the other one. In other words, there must exist a tensor R αβ α β (λ, µ) which acts on the virtual indices which switches the local tensors of O(λ) with O(µ) (see Figure 11). Indeed, if there exists a tensor R with that property, we can put I = R 1 R somewhere on the virtual level, pull through the R matrix all along the MPO with periodic boundary conditions, and then let it annihilate again as RR 1 = I. This way of identifying two different MPOs by pulling through a different tensor R is also the way in which the nonequilibrium problems will be solved, and also plays a crucial role in the numerical treatment of diagonalizing MPOs. The essence of the algebraic Bethe ansatz now consists of realizing that those R-matrices have to satisfy an associativity condition, and that the corresponding algebraic equations yield a very rigid framework for possible solutions. The associativity property of the R- matrices follows from the fact that there are two different ways in which we can reorder three 22 Haegeman and Verstraete

23 Figure 11: R-matrix as the gauge transformation switching O(λ) and O(µ). Figure 12: Yang-Baxter equations as associativity conditions for the R-matrices in the algebraic Bethe ansatz. We use the notation R for the inverse of R. commuting MPOs in a local way (see Figure 12). The two triples of R-matrix contractions hence have to be equal to each other, i.e. R αβ α β (µ, ν)r β γ β γ (λ, ν)r α β α β (λ, ν) = R βγ β γ (λ, µ)rαβ α β (λ, ν)rβ γ β γ (µ, ν) where Einstein summation convention is assumed. These equations are called the Yang- Baxter equations [85, 86]. The solutions of these Yang-Baxter equations are very restrictive, and effectively all integrable models in statistical physics can be constructed from those solutions. It turns out that the defining property of the R-matrix, as illustrated in Figure 11, is of the exact same form as this Yang-Baxter equation. Given such a set of solutions of the Yang-Baxter equations {R(λ, µ)}, it is now easy to construct a specific set of MPOs {O(λ)} satisfying the required commutativity relations. This is the so-called fundamental representation which is obtained by choosing A ij αβ (λ) = Rαi jβ(λ, 0). Let us now illustrate how to proceed for finding the eigenvectors of this set of commuting MPOs for the simplest nontrivial model, the XXX model. We will follow the steps as explained in the book of Korepin, Bogoliubov and Izergin [45]: 1. The simplest solution of the Yang-Baxter is given by the R-matrices f(µ, λ) g(µ, λ) 1 0 R(λ, µ) = 0 1 g(µ, λ) f(µ, λ) with g(µ, λ) = i/(µ λ) and f(µ, λ) = 1 + g(µ, λ) for the specific XXX model. The corresponding fundamental representation is of the form A ij αβ (λ) = iλδijδ αβ + δ iαδ jβ (see Figure 13) and hereafter we have constructed all MPOs with those elements. Exact and Computational Methods for Matrix Product Operators 23

24 Figure 13: Local tensors of the matrix product operator in the XXX model. 2. It is now possible to construct 4 different families of MPOs by varying the boundary conditions M: [ ] [ ] [ ] [ ] We will call the 4 corresponding MPOs A(λ), B(λ), C(λ), D(λ), where T = A + D corresponds to closing with the identity matrix M = 1. As can easily be checked, the state Ω = is an eigenstate of A(λ), C(λ), D(λ) with eigenvalues a(λ) = (1 + iλ) N, 0, d(λ) = (iλ) N, respectively. 3. The Yang-Baxter relations impose the following commutation relations on those MPOs: We can now check that [A(λ), A(µ)] = 0 = [B(λ), B(µ)] = [D(λ), D(µ)], A(µ)B(λ) = f(µ, λ)b(λ)a(µ) g(µ, λ)b(µ)a(λ), D(µ)B(λ) = f(λ, µ)b(λ)d(µ) g(λ, µ)b(µ)d(λ). A(µ) i B(λ i) Ω = Λ i B(λ i)a(µ) Ω + n Λ nb(µ) j n B(λ j)a(λ n) Ω, D(µ) i B(λ i) Ω = Λ i B(λ i)d(µ) Ω + n Λ nb(µ) j n B(λ j)d(λ n) Ω, The coefficients for Λ and Λ are easily obtained, as there is only one way by which A(µ) and D(µ) were commuted to the very right: Λ = i Λ = i f(µ, λ i), f(λ i, µ) The equations for Λ n look much more complicated, as there are in principle exponentially many ways (2 n 1 ) by which A(λ n) could be commuted to the very right. Note that there is only one way by which the term with A(λ 1) can be obtained, and the coefficient in front of the corresponding term is thus easily obtained. Without loss of generality, we could however have reordered all commuting B(λ i), and hence could have obtained all other terms in the same way. This leads to the following equations: Λ n = g(λ n, µ) j n f(λ n, λ j), Λ n = g(µ, λ n) j n f(λ j, λ n). 24 Haegeman and Verstraete

25 Figure 14: The logarithmic derivative of the 6-vertex matrix product operator yields the Heisenberg spin 1/2 Hamiltonian. 4. If we can now find solutions {λ i} for which a(λ n)λ n + d(λ n) Λ n vanishes for all n, then we clearly have found an eigenstate of T (µ) = A(µ) + D(µ). It turns out that the equation a(λ n)λ n + d(λ n) Λ n = 0 exactly corresponds to the well known Bethe equations which also appear in the coordinate Bethe ansatz. For many cases, and in particular for the case considered here, it is known how to obtain a complete set of solutions of those nonlinear equations, and every different solution corresponds to an orthogonal eigenstate i B(λi) Ω with eigenvalue a(µ)λ + d(µ) Λ. B(λ i) can hence be interpreted as a creation operator, creating a particle with (quasi-)momentum λ i and energy f(µ, λ i). It is interesting to note that the algebraic Bethe ansatz yields a matrix product ansatz for all eigenstates, albeit one where the dimension scales exponentially with the number of particles and/or applications of B operators [87]. Indeed, e.g. the spin zero eigenstates on a chain with N sites are obtained by multiplying N/2 MPOs B(λ) with bond dimension 2 with the vacuum, yielding an MPS with bond dimension 2 N/2. By constructing an algebra of MPOs, we have hence shown that is is possible to diagonalize a set of commuting MPOs T (µ). Most of the integrable models in statistical physics can be solved along those lines, and there is a very rich mathematical literature dedicated to working out further intriguing algebraic properties of those equations. It turns out that there is a beautiful connection between the above XXX classical statistical model and the Heisenberg antiferromagnet. Figure 14 illustrates the fact that the logarithmic derivative of the MPO T (λ) at λ = 0 H = T 1 d T (λ) dλ λ=0 is precisely the Heisenberg model. As all T (λ) commute, this automatically implies that they share the same eigenstates. The leading eigenvector of T (λ) corresponds to the ground state of the Heisenberg antiferromagnet for 1 λ 0. From the point of view of numerics, the optimal choice seems to be λ = 1/2, as this value maximizes the gap of the transfer matrix. A similar construction can in principle be done for the one-dimensional Hubbard model, which was also shown to be integrable by Lieb and Wu [88]. Numerical MPS method can readily be applied to this model, which allows to calculate correlation and spectral functions The coordinate Bethe ansatz. Historically, the Bethe ansatz was first formulated in terms of first quantization [84] and called the coordinate Bethe ansatz. We will show that this formulation can also immediately be converted into an MPS description with exponential bond dimension. It will turn out that this MPS description is equivalent to the MPS description emerging from the algebraic approach up to a gauge transform [80]. Let us consider the Heisenberg spin 1/2 ferromagnet for simplicity, although similar Exact and Computational Methods for Matrix Product Operators 25

26 results can easily be derived for more complicated integrable models. Bethe assumed that the empty vacuum state is all spins down, and wrote down an ansatz for all eigenstates of a given magnetization M with spins up at positions n 1, n 2, n M : ψ (n 1, n 2,, n M ) = P ( ) M exp i n jξ Pj exp i 1 θ ( ) ξ Pj, ξ Pk 2 j=1 Here P is the set of all permutation operators, the ξ i are the spectral parameters, ξ Pj is the j th spectral parameter after the permutation, and θ (ξ i, ξ j) is the antisymmetric logarithm of the scattering matrix which can be determined from the two-particle problem. The spectral parameters are obtained by imposing that this ansatz is an actual eigenstate, leading again to the Bethe equations. It is remarkably simple to write an MPS in the second quantized form which is equivalent to ψ. Let us first consider the case of 1 particle, which has the form ψ = 1 N N j=1 e iξj ψ j Ω and which can readily be identified with the MPS A 0 (ξ) = D(ξ), A 1 = J, M = V : D(ξ) = [ ] e iξ J = [ ] 0 e iξ 0 0 j<k [ ] 0 0 V = 1 0 The 2-particle case with spectral parameters ξ 1, ξ 2 can then be obtained by introducing A 0 = i D (ξi), M = i V and A1 = B 1 + B 2 with B 1 = B 2 = [ e i(ξ θ(ξ 1,ξ 2 )) 0 0 e i 1 2 θ(ξ 2,ξ 1 ) [ ] [ ] 0 e iξ 2 e iξ ] [ ] 0 e iξ 1, 0 0 The important property of those matrices is the fact that B i ( A 0 )k B j B i = 0, and the boundary conditions then imply that both of them have to appear exactly once in the MPS wavefunction. If B 1 appears to the left of B 2, then the scattering phase θ (ξ 1, ξ 2) /2 is introduced, and in the other case the minus θ (ξ 2, ξ 1) /2 is introduced. This is clearly a MPS of bond dimension 4, and we can readily proceed to the M-particle case: A 0 = [ ] e iξ j 0 M A 1 = B j M = 0 1 j j=1 j [ ] [ M e i(ξ k+ 1 2 θ(ξ j,ξ k)) 0 0 e iξ j B j = 0 e i 1 2 θ(ξ k,ξ j) 0 0 k=j+1 [ ] [ ] e iξ k ] j 1 We can indeed check that this ordering reproduces the right phases. The resulting MPS has bond dimension 2 M. This proves that an MPS is able to encode the permutation properties of Bethe ansatz wavefunctions very easily. Note also that the case of free fermions can readily be obtained by imposing θ(λ i, λ j) = π, which makes the wavefunction antisymmetric. The k=1 26 Haegeman and Verstraete

27 matrices involved in the MPS satisfy the following commutation relations, which are related to the Zamalodchikov algebra of creation operators in a 2-dimensional quantum field theory [89]: B ja 0 = e iξ j A 0 B j (B j) 2 = 0 B jb k = e iθ(ξ j,ξ k) Bk B j Those relations were first discovered by Alcaraz and Lazo [90, 91], whose goal was to find a matrix product ansatz for eigenstates of integrable systems. They showed that the condition of an MPS to be an eigenstate of the Heisenberg Hamiltonian is equivalent to those relations, and henceforth managed to find representations of this algebra. At first sight, this MPS solution looks distinct from the one obtained from the algebraic Bethe ansatz. However, Katsura and Maruyama constructed a gauge transformation [80] which transforms both into each other: The fundamental theorem is at work again and allows going form a first quantized description to a second quantized one! We illustrate this utilizing notation used in the previous section. It can readily be checked that A 0 in the MPS description in the algebraic Bethe is upper diagonal. An upper diagonal matrix can be diagonalized by a similarity transformation with an upper triangular matrix Q with 1s on the diagonal, and this leads to QA 0 Q 1 = [ ] iλ j iλ j j Applying the same similarity transform on A 1 leads to QA 1 Q 1 = i Bi with B j = k>j [ ] [ (1 + iλ k )f(λ k, λ j) iλ k f(λ j, λ k ) 1 0 ] k<j [ ] iλ k iλ k We can now identify the corresponding terms as a function of {λ i} with the terms obtained in the coordinate Bethe ansatz as a function of {ξ i}. As an example, let s go back to the 2-dimensional spin ice problem of Lieb, for which the entropy was given by the leading eigenvalue of the MPO A ij αβ = { 1, i + j + α + β = 2 0, i + j + α + β 2 By multiplying the vertical legs of the MPO by σ x Pauli operators, this MPO tensor is of the form A which is precisely of the same form as the R-matrices introduced in section We can hence imbed this MPO in a 1-parameter family of Bethe ansatz integrable models, and find the exact eigenvectors either by means of the algebraic or coordinate Bethe ansatz. The resulting eigenvalue per site is given by Lieb s square ice constant 8 3/9. Exact and Computational Methods for Matrix Product Operators 27

28 Figure 15: operators. The fundamental theorem at work on a closed algebra of matrix product Figure 16: Associativity conditions for the fusion tensors X Discrete MPO algebras and tensor fusion categories The section on the algebraic Bethe ansatz already demonstrated that MPOs can exhibit a very rich algebraic structure. The central ingredient of the algebraic Bethe ansatz arose from the fundamental theorem of MPOs: the global commutativity of an algebra of MPOs leads to the existence of a local R-matrix with nontrivial algebraic conditions which follow from associativity. In the case of the Bethe ansatz, the set of MPOs was characterized by a continuous spectral parameter λ. What about algebras of MPOs with a discrete label? Such algebraic constructions exactly lead to representations of tensor fusion categories [92], which form the mathematical basis for describing topological order and theories exhibiting anyons in two spatial dimensions. As found out by Drinfield and Jimbo, such constructions are very much related to the Yang Baxter equations, and the discrete structure is obtained by taking the limiting case of the spectral parameters being equal to infinity. Their work gave rise to the field of quantum groups. The logic to find solutions parallels the logic followed in the previous section: solutions of the associativity conditions of the gauge transforms will allow us to construct fundamental representations. First of all, we want to construct a discrete set of injective MPOs {O a} which form a closed algebra of MPOs with structure factors independent of the size of the MPOs: O ao b = NabO c c c The tensor Nab c consists of integers and encodes the so-called fusion rules, i.e. the different ways in which the MPOs can fuse into other ones. The fundamental theorem of MPOs then implies that there must be a gauge transform Xabµ c (where µ stands for a possible degeneracy) which decomposes the joint MPO O ao b into a direct sum of blocks (see Figure 15). The X-tensors have to satisfy the associativity condition depicted in Figure 16. Due to the fundamental theorem and the injectivity of the MPOs, this implies the existence of a 28 Haegeman and Verstraete